Basic Fluid Mechanics

### **Detailed Step-by-Step Solution (GATE AE 2025 Q.41)** **1. Intuitive Physical Setup & Given Parameters**: We are asked to evaluate the standard calculus limit: $L = \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2}$ **2. Step-by-Step Mathematical Derivation**: 1. **Direct Evaluation (Indeterminate Form)**: At $x = 0$, the expression gives $\frac{1 - \cos(0)}{0} = \frac{1 - 1}{0} = \frac{0}{0}$ (indeterminate form). 2. **Method 1: Trigonometric Half-Angle Identity**: Using $1 - \cos(2x) = 2\sin^2(x)$: $L = \lim_{x \to 0} \frac{2\sin^2(x)}{x^2} = 2 \left(\lim_{x \to 0} \frac{\sin x}{x}\right)^2 = 2(1)^2 = 2$ 3. **Method 2: L'Hôpital's Rule**: Differentiating numerator and denominator: $L = \lim_{x \to 0} \frac{2\sin(2x)}{2x} = \lim_{x \to 0} \frac{\sin(2x)}{x} = 2 \lim_{x \to 0} \frac{\sin(2x)}{2x} = 2(1) = 2$ 4. **Method 3: Taylor Series Expansion**: $\cos(2x) = 1 - \frac{(2x)^2}{2!} + O(x^4) = 1 - 2x^2 + O(x^4)$ $1 - \cos(2x) = 2x^2 - O(x^4) \implies \lim_{x \to 0} \frac{2x^2}{x^2} = 2$ **3. Comprehensive Options & Distractor Analysis**: - Numerical Answer Type (NAT) Question: Integer expected. - Exact Integer Value: **2**. - Official Key: `2` (Range: `[2, 2]`). **4. Official Master Key & Exam Takeaway**: - **Official Answer Key**: **2.0** (Acceptable Range: `[2, 2]`) - **Exam Takeaway**: Always remember the standard asymptotic expansion $1 - \cos(kx) \approx \frac{k^2 x^2}{2}$ as $x \to 0$. Here with $k = 2$, $\frac{k^2}{2} = \frac{4}{2} = 2$.